| Exam Board | OCR MEI |
|---|---|
| Module | S1 (Statistics 1) |
| Marks | 7 |
| Paper | Download PDF ↗ |
| Mark scheme | Download PDF ↗ |
| Topic | Discrete Probability Distributions |
| Type | Simple algebraic expression for P(X=x) |
| Difficulty | Moderate -0.8 This is a straightforward probability distribution question requiring basic techniques: summing probabilities to find k (simple algebra with small numbers), then applying standard expectation and variance formulas. All steps are routine calculations with no conceptual challenges or problem-solving insight needed. |
| Spec | 5.02a Discrete probability distributions: general5.02b Expectation and variance: discrete random variables |
| Answer | Marks | Guidance |
|---|---|---|
| \(P(X=1)=7k,\ P(X=2)=12k,\ P(X=3)=15k,\ P(X=4)=16k\) | M1 | for addition of four multiples of \(k\) |
| \(50k = 1\) so \(k = \frac{1}{50}\) | A1 | ANSWER GIVEN |
| Answer | Marks | Guidance |
|---|---|---|
| \(E(X) = 1\times7k + 2\times12k + 3\times15k + 4\times16k = 140k = 2.8\) | M1 | for \(\Sigma xp\) (at least 3 terms correct) |
| A1 CAO | ||
| \(\text{Var}(X) = 1\times7k + 4\times12k + 9\times15k + 16\times16k - 7.84\) | M1 | \(\Sigma x^2p\) (at least 3 terms correct) |
| \(= 8.92 - 7.84 = 1.08\) | M1dep | for their \(E(X)^2\), provided \(\text{Var}(X) > 0\) |
| A1 FT | their \(E(X)\) |
## Question 5:
### Part (i)
| $P(X=1)=7k,\ P(X=2)=12k,\ P(X=3)=15k,\ P(X=4)=16k$ | M1 | for addition of four multiples of $k$ |
| $50k = 1$ so $k = \frac{1}{50}$ | A1 | **ANSWER GIVEN** |
### Part (ii)
| $E(X) = 1\times7k + 2\times12k + 3\times15k + 4\times16k = 140k = 2.8$ | M1 | for $\Sigma xp$ (at least 3 terms correct) |
| | A1 CAO | |
| $\text{Var}(X) = 1\times7k + 4\times12k + 9\times15k + 16\times16k - 7.84$ | M1 | $\Sigma x^2p$ (at least 3 terms correct) |
| $= 8.92 - 7.84 = 1.08$ | M1dep | for their $E(X)^2$, provided $\text{Var}(X) > 0$ |
| | A1 FT | their $E(X)$ |5 The score, $X$, obtained on a given throw of a biased, four-faced die is given by the probability distribution
$$\mathrm { P } ( X = r ) = k r ( 8 - r ) \text { for } r = 1,2,3,4 .$$
(i) Show that $k = \frac { 1 } { 50 }$.\\
(ii) Calculate $\mathrm { E } ( X )$ and $\operatorname { Var } ( X )$.
\hfill \mbox{\textit{OCR MEI S1 Q5 [7]}}